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CHUaNG 2

NGHlltM TUAN HoAN CUA PHUONG TRINH
VI PHAN TmJONG
2.1. Dfnh ly chinh :
La"y 1=[0, 1], va f: I x RD~ RD,

thoa nhii'ng di~u ki~n caratheodory, va ky hi~u x la chu§:nEuclide cua
I

I

x E RD,va (x, y) la tich va htidng cua x va y.
Trong chtiang nay, chung ta se chung minh st! t6n tC;iinghi~m cho bai
tmin.
(2.1)

x1(t)= f(t,x(t))

, tEl

{ x(o) = x (1)

nhii'ng nghi~m nay se dtiqc gQila 1 tu~n hoan.
Chung ta ky hi~u X la khang gian con cua C (1, RD)ma nhung ph~n
tll'cua no thoa di~u ki~n thu hai trong (2.1) vdi chu§:nd~u thtiong dung la
Ixl

0

= maxlx(t)\

tEl

'

Z =L 1 (I, RD),vdi chu§:nthtiong dung la
IIxiiI

= fixet)ldt
I

va domL la khang gian con cua X, ma nhung ph~n tii' cua no la lien t\lC
tuy~t dot
Anh x~ L va N l~n ltiqt dtiQCxac dinh tren domL va X, bdi
(L.x)(t)

= x'(t),

(Nx) (t)

=f(t, x(t))

Vdi tEl, thl L va N la'ynhung gia tri trong Z, va bai loan (2.1) ttiang
duang vdi vi~c giai phuong trlnh thu gQn
Lx = Nx

25


ngoai fa, tll slj ma ta trong chuang 1,

= {x E dam

KerL

IrnL

d

= {z EZ

day Px

L : xCi) = x(o), Vt E I}

: fZ(I)dl

= x(o),

Qx

= ImP

=o} = KerQ

= fz(t)dt
I

VI the'L la anh x']. Fredholm vdi chi sO'zero va do tinh chat 1.5, ta co
N la L - hoan loan lien tl;lCtrong X.
Ta c~n m(>tb6 d€ d€ chung minh slj t6n t'].inghi~m.

B6 d~ 2.1 :

Cho r > 0 va V E Cl(Rn, R), thoa man
V' (x) * 0, vdi x
I

I

=r

?

d day V' la gradient cua V, va Iffy
G :X

(Gx) (t)
va H

=L -

Z, duQcxac dinh bdi

~

=-

V' (x(t)),

tEl

G, vdi X, Z va L duQCxac dinh nhutren.

Thl H E CL(B(r)) va !DL(H,B(r))! =\Do(V',B(r))\
Chung minh
Ta xet anh x'].
H :XxI~Z
(x, A) H Lx - AGX- (1 - A) QGx
Vdi Q duQc xac dtnh d ireD.
Thl H la L - hoan toaD lien tl;lcireD X x I.
Ne'u (x, A) E X x I, saG cho H (x, A) =0, thl x la lien wc tuy~t d6i 1 - tu~n

hoan va

26


(2.2)x'(t)=-

AV'(X(t))-(l-A)fV'(X(S))ds,
I

'\itEI

Do do x' lien wc tren I va liy tich vo huang hai v~ (2.2) vai x'(t),
tich phan tren I va dung Hnhchit 1 - tu~n ho~mcua x, ta duQc.
fix' (012dt = 0
I
Vi v~y x(t)
x'(o)

=x(o),

'\it E I, bdi vi

= -AV'(x(o))-

(1- A)fV'(x(s))ds
I

Bdi (2.2), x(o) thoa phudng trlnh V'(x(o))

=0

f)i~u nay guy ra ding
\xlo

= Ix(o)\"*

r

Theo tinh chit bit bi~n d6ng luau cua ly thuy~t b~c, ta co :
DL(H,B(r))

= DL(H(.,l),B(r)) = DL(H(.,O), B(r))
= DL(L - QG,B(r))

Nhung QG : X ~ ImQ, vai

z= ImL~ImQ
Vi v~y, bdi tinh chit (1.13), ta co

\DL(L - QG,B(r))1 = \Do(-QGKerL,B(r)nKerL)1
=IDo(V',B(r))!

d

day B(r) duQc ky hi~u qua c~u tam 0, ban kinh r. B6 d~ duQc

chung minh.
Bay giC1ta chung minh dinh ly chinh cua chudng.
Binh Iy 2.2.
Gia sa ding nhung di~u ki~n sail day xay ra

27


(i) co mQt V E C1 (Rn, R+), vdi :
Vex) ~ + 00, ne'u

I

x

I

~

00

va a ELI (I, R+), thoa man

s aCt)

(2.3) (V' (x),f(t,x»)

Vdi mQi x ERn, mQi tEl
(ii) T6n t~i r > 0 va W E C1 (Rn \ B(r), R), sao cho
(V' (x), w' (x» > 0
VdimQix,

Ixl ~r,va

(2.4) f(W'( x(t)),f( t,x(t)))dt
I

s0

Vdi mQi anh x~ lien tl,lctuy~t d6i 1 - tugn hoan
x : I ~ Rll, vdi minlx(t)1~ r
tel
Thl bai toan (2.1) co it nha't IDQtnghi~ID

ChUng mink

.

Ta mu6n ap dl,lngdinh 19 (1.17) vdi
F

=L -

N va H = L - G

nhtt trong b6 d~ (2.1) va tinh cha't cQng tinh, cling vdi V' (x)
x E Rn vdi

I

x ~r
I

"*

0, cho IDQi

.

H eCL(B(p») vdi mQip ~ r
£)gu tien chung ta chI ra ding nhii'ng nghi~m 1 - tugn hoan cua hQ
nhii'ng phu'dng trlnh.
x' (t)

=-

(1 - A) V' (x(t» + Af (t, x(t», tEl,

AE10,1[ la mQt ti~n bi

ch~n.
Ne'u di~u nay kh6ng Kay fa, se co mQt day (An)nEN*vdi An E ]O,l[

28


va mQt day

(xn)n EN*, vdi IXnlo2 n va xn la mQtnghi~m 1 - tu~n hoan cua

(2.5): x~(t)=-(l-An)V'(Xn(t))+Anf(t,xn(t)),

tEl, n EN*

Ngoai fa, vdi m6i tEl, n E N*, dung di~u ki~n (i)
(d/dt) V (Xn(t))=(V'(Xn(t)),X~(t))
= -(1- An).IV'(xn (t))12+ An(V' (Xn(t)), f( t, xn (t))) :s;aCt)
Md fQng Xnva a fa R bdi 1 - tu~n llOan, ta du'Qc vdi m6i T E R va
m6i t E [T, T + 1]
v( xn (t)):S;V(xn (T)) + Jf
r a(s)ds
Do tinh 1 - tu~n hoan cua Xn,di~u nay suy fa ding
(2.6)

max V( Xn(t)) :s;mill V( xn (t)) + Ilal!l
tEl
tEl

Bay giO, n€u tn E I sao cho
\Xn (tn)1 = maxlxn (t)1= \Xnl 2 n

tEl

0

Ta co
max V(Xn(t)) 2 V(xn (tn))
tEl
va bdi di~u ki~n (i)
V(Xn(tn))~oo

n€u n ~oo.

Ngoai fa bdi (2.6)
mill V(xn(t))~
tEl

00 n€u n ~ 00

Hoan loan tu'dng tt,I'suy fa ding
minlxn(t)1 ~ 00ne"u n ~ 00
tEl

29


La'y nl E N* sao cho, voi m6i n ~ nl
minlxn (t)j ~ r
tEl
Bi~u sail duQc suy ra tti' (2.5)
(d / dt)W( xn (t)) = -(1- An)(V'(xn(t)),W'( xn(t))) + An(f( t,xn (t)),W,(xn(t)))

Va VIv~y, bdi tinh 1 - tuftn hoan cua Xn,va voi n ~ nl, ta suy ra tti'

(ii)
0 = -(1- An) f(V'( xn (t)),W'(xn (t)))dt + An J(f(t,xn(t)), W'(xn (t)))dt < 0
I
I
di~u nay mall thu~n.
VI v~y nhG'ng nghi~m cua hQ nhG'ngphuong trlnh 1a ti~n bi ch~n, vOi
ti~n bi ch~n duQc ky hi~u bdi p. Chung ta co th€ chQn p sao chop ~ r.

Bay gio dung b6 d~ (2.1), gia dinh (i) va tinh cha't (1.15), chung ta co
IDL(H,B(p))!

= IDo(V', V(p)~ = 1

Va bdi dinh 1y (1.17), dinh 1y duQc chung minh.
Binb Iy 2.2' :
Gia sa nhG'ng di~u ki~n san day Kay ra
(i') T6n t~i V cl (Rn, Rr), voi
Vex) ~ + 00 ne'u

Va a

I

x

I

~

00

ELl (I, R+) sao cho

(2.3') (V' (x), f(t, x)) ~ -aCt)
voi mQi x ERn, mQi tEl
(ii') T6n t~i r > 0 va W E Cl (Rn\B(r), R), saG cho
.

(V'(x), Wi(x)) > 0-, voi mQi x, Ix! ~rva

30


(2.4')

f(W'(x(t)),f(t,x(t)))dt?::
I

0

Vai mQi X : 1 ~ Rn lien t~c tuy~t d6i 1 - tuftn hO~lllvai minlx(t)l?::r.
tEl
ChUng minh
D6i bi~n, bftng cach d~t t = 1 - T
Vai mQi tEl

co TEL

Luc niiy(2.3')
(2.3 ')

vii (2.4') trd thiinh

(- V'(x(1- T)),f(l- T,x(l- T)))?::-a(l- T)
~

(2.4 )

(V'(x(1- T)),f(l- T,x(l- T))) ~ a(l- T)

SI( -W'(x(1-

T)), f(l- T, x(l- T))) d(1- T) ?::0

Bftng cach d6i bi~n mQt Iftnnua 1 - T = 't
Ta co (2.3') vii (2.4') trd thiinh
(V'(x('t) ),f( 't,x( 't))) ~ a('t)
f(W'(X('t)),f('t,X('t))}t't ~ 0
I
Ap d~ng dinh 19 (2.2), dinh 19 duQc chung minh.
2.2.

Ung

d\lng cua dinh Iy 2.2.

Trong phftn niiy, chung ta se cho mQt viii ung d~ng thu vi cua djnh 19
2.2, vai sl;l'll;l'achQn d~c bi~t. Ung d~ng dftu lien Iii cho phuong trinh vo
huang (n = I).

H~ qua 2.1 :

Gia sa n = 1 vii a.e tEl,

f(t,.) Iii khong tang. Thl biii loan (2.1) co
00

mQt nghi~m n~u vii chi n~u t5n t~i y E L (I, R) sao cho

31


(2.7)

J f( t,y(t) )dt = 0
I

Chung minh
* Di~u kit%nc~n :
Ta Iffy Y Hi mQt nghit%mcua (2.1), suy fa

Jf( t,yet))dt = Jy' (t)dt = y(l) - yea) = 0
I

I

* Di~u kit%ndu :
VI f(t,.) la khong tang nen ta co, voi mQi x E R va a.e tEl,
xf(t,x) ~ x f(t,O) ~ \f(t,O)I.(lxl + 1)

Do do di~u kit%n(i) cua dinh ly (2.2) thoa man voi

V(x)

aCt)= f(t, 0)
I

=G)

(

1

( ul

1-1

+1) du

I.

Bdi VI Vex) ~ +

00

ne'u x
I

I

~

00

Va a ELI (I, R+), saG cho
(V' (x),f(t, x)) = x(\xl + 1)-I.f(t, x)
= (Ix!+ lr1 x f(t,x)
~ (lx\+ lr1\f(t,0)1.(lx\

+ 1)

~ \f(t,o)1 = aCt)

Ta Iffy x E domL la ph~n tii'tily Y saG cho :
minlx(t)1~ Ilyll = f
tel

32


Dung tinh don di~u khong tang cua f(t,.), ta co a.e tEl,
x(t) . f(t, x(t))

s x(t) . f(t, yet))

Suy fa

f X(t).f(x(t)1t, x(t») dt s
I

I

f x(t)

I Ix(t)!

.f( t, yet) )dt

S:I: ff(t,y(t))dt=O
I
2

1 x
Nhu'ng la'y w(x)

= 2 12

--

1

u 2 du

x
W'(x)

=~

Suy fa

WEe

1

(R \B(r), R)

It 1:(°,vdi

.Ix I ~r

Va Jw'(
x( t) );f(t, x(t) )dt = IJ1:~:~l"f(
I

t,x(tJ)dt,;;

V'(x).

W'(x) = (Ixl+

Vdi x : I

~

0

R lien tl;lctuy~t d6i 1 - tu~n ho~m.VI v~y di~u ki~n (ii)

cua dinh ly (2.2) xay fa, theo dinh ly nay, h~ qua du'Qcchang minh.

H~ qua 2.1' :

Gia sa n = 1 va a.e tEl, f (t,.) la khong giam. Thl bai tmln (2.1) co
00

mQt nghi~m ne'u va chI t6n t~i y E L (1, R) sao cho
(2.7')

ff(t,y(t»)dt=O
I

-

33


Chung minh
* Di6u ki~n cftn :
Ta la'y y la mQt nghi~m cua (2.1), suy fa
Jf(t,y(t»)dt=
I

Jy'(t)dt=y(1)-y(O)=O
I

* Di6u ki~n du :
Vi f(t,.) la khong giam, nen ta co, vdi m6i x E R va a.e t E I,
x.f(t,x) ~ x f(t, 0) ~ - \f(t,O)I(lx!+ 1)

Do do di6u ki~n (i') cua dinh ly (2.2') thoa man, vdi

-1
1

Vex)

~2

!

= (2) r l u2 + 1J

Bdi vi Vex) ~ +

00

du. aCt) = !f(t,O)1

ne'u x
I

I

~

00

va a ELI (I, R+)

Sao cho :
(V' (x),f(t,x») = x(lxl+ lr1f(t,x)

= (Ix! + 1)-lXf(t,x) ~ (lxl+ lr1[-\f(t,O)I(lxl + 1)]
~ -\f(t,O)\

= -aCt)

Ta la'y x E domL la phftn tii'tily Y sao cho
Minlx(t)1 ~ Ilylloo= f

Ta dung tinh don di~u khong giam cua f(t,.), a.e t E I,
xCi) . f(t, xCi»~~ xCi) f(t, yet)

34


Suy ra
f(t,y(t))dt
J X(t).f(t,X(t))dt;::: JX(t)
IX(t)1
IX(t)1
I

I

I

;::::tJf(t,y(t))dt=O
I

1 x2
Nhu'ng la'y W(x)

_!

=2 £2 u

W' (x)

2du

= x(t)
IX(t)1

1

Suy ra WEe

V' (X)

Vii

(R\B(r), R)

r

. W' (X) = x(lxl+ 1

11~ > 0, vdi

I

x

I

;?:

r

JW'(x(t)).f(t,x(t))dt
= J1:i:~I.f(t'X(t))dt
z 0,
I
I
-

Vdi x : I

~

R lien t~c tuy<%td6i 1 - tuftn hoan. Do do di~u ki<%n
(ii')

cua dinh ly (2.2') thoa man, ap d~ng dinh ly nay, h<%
qua du'Qcchung minh,
H~ qua 2.2 :
Gia stt t6n t(;l.ir > 0 va a E Ll (1, R+), sao cho
(2,8)

(x, f(t, x)) ~ aCt) (lx12+ 1), a.e t EI

Vdi mQi x E Rll, va
(2.9)

J(X(t),f(t,x(t)))dt~O
I

, vdi mQi x E domL

minlx(t)l;::: r
teI
Thl bai loan (2.1) co it nha't mQt nghi<%m

35


Chung minh
2

1 xl
(u+ 1) 1du,
Lay Vex) = -~:
2t
,.:'

W(x) =-

x
11

2

2

Suy fa

(

2

-1

)

V'(x) = x Ixl + 1

,

Ta ki~m tra di8u ki~n (i) va (ii)

W'(x) = x

cua dinh

(2.2).

1y

Ta c6 V E C1 (Rn, R;.), vdi
Vex) ~ +

(V'(x), f(t, x)) = (x(lxl2

00

ne'u

I

x

I

~

00

+ It,f(I,X))

~ a(t{lxI2 + 1)(lxl2 + 1)-1 = aCt)
Vdi mQi x E Rn, a.e tEL
Do d6 di8u ki~n (i) cua dinh 1y (2.2) thoa man:

M~tkhac

W E C1(Rn\B(r), R)

(V'(x), W'(x)) = (x(JxI2+It,~

>0

V di mQi x, x 12 f , va
I

f(W'( x(t) ),f( t,x(t) ))dt = J(X(t),f( t,x(t) ))dt ~ 0
I
I
V di mQi x E domL
minlx(t)12 r
tel
Do d6 di8u ki~n (ii) cua dinh 1y (2.2) thoa man.

36


V~y bai tmin (2.1) co it nha't mQt nghi~m
H~ qua 2.2' :
Gia su t6n tq.ir > 0 va a ELI (I, R+), sao cho
(2.8') (Xf(t,X))2-a(t)(lxI2

+1), a.et E1, va mQi x ERn, va

(2.9') f(X(t),f(t,x(0))dt20,
I

vdi mQi x E domL
minjx( 012 r
tel

Thl bai tmin (2.1) co it nha't mQt nghi~m
Chung minh

1

,2
, W(x) = ~
2

La'y Vex) = - x (u +1)-ldu
2

i

12

I

Suy ra V' (x) = x(!xj2+ 1)-1 , W'(x)

=x

Ta se ki~m tra di~u ki~n (i') va (ii') cua dinh ly (2.2').
Ta co V E C1(Rn,R+), vdi
Vex) ~ +

00

,

n6u x I ~
I

00

(V'Cx),f(t, x)) = ( x(lxl2 + 0-1 ,fCI,X)) ~ -aCtJ(lxI2 + 1)(jxj2 +
2-

aCt)

a.e t E I va mQix ERn.

Do do di~u ki~n (i') cua dinh 1y (2.2') thoa man.
M~t khac

W E C1 (Rn \ B(r), R)

(V'(x), W'(x» =

(x(lxI2+ It ,X)>0
37

r


Voi mQi x, x 12 r
I

Va f(W'(x(t)),f(t,x(t))}it = f(X(t),f(t,X(t)))dt 2 0
I
I
Voi mQi x E domL, minlx(t)12 r
tel

Do do di~u ki~n (ii') cua dinh 1y (2.2') thoa man.
V~y bai tmin (2.1) co it nha't mQt nghi~m.

H~ qua 2.3:
Gia sa co r > 0 sao cho
(2.10)

Voi x
I

(x, f(t, x)) ~ 0,

a.e tEl

va mQi x E Rn

=r

I

Thl bai tmin (2.1) co it nha't mQt nghi~m
ChUng minh

Ta dinh nghIa
g : 1 x Rn
get, x)

g(t, x)

R 0, bdi

~

= f(t, x)

neu

I

x ~ r,
I

= + -1:1)x+r(t'I:10,
neu

I

x

I

;::

r

Bdi sl,txay dl,tng tren, g cling thoa nhii'ng di~u ki~n caratheodory nhtt
f va trung voi f tren x ~ r.
I

I

Ngoai fa, neu a ELI (I, R+) thoa man.
I

f(t, x) ~ aCt), a.e tEl
I

(x, g(t,x))

va mQi x ERn, ydi x ~ r, thl
I

I

~ aCt) (lxf + 1), vdi mQi x E Ro, a. et E I.

Do do di~u ki~n (2.8) cua h~ qua (2.2) thoa man cho g

38


Neu x E domL thoa man minlx(t)1~ r.
tel
Taco
(2.11)
(x(t),g( t,x(t»))

= -( 1-lx~t)Jlx(tf
~- 1-~

(

Ix(t)!J

+ ( x(t), f( 1, Ix~t)1

X(t)]

J

lx(t)12~O

La'y tich phan tren I, di~u ki~n (2.9) ~ua h~ qua (2.2) thoa man cho g
va bai loan.
(2.12)

x1(t) = g(t,x(t») ,t EI

{ x(o) = x(1)
Co it nha't mQt nghi~m. Ta chung minh nghi~m x nay thoa man
Ix(t)\~ r ,tEl

Neu chung minh duQcdi~u ki~n nay, thl nghi~m cua (2.12) cling la
nghi~m cua (2.1).
.

Neil x la mQtnghi~m cua (2.12), thl bai (2.11)
Ta co a.e tEl sac cho x (t) > r
I

(2.13)

G)(d/

I

dtJlx(tJl2 = (x(tJ,g{ t,x(t)))

~_

(

1- ~

Ix(t)1)

lx(t)12~ 0

Do do nghi~m nay khong th~ la 1 - tuftn bean va thoa man x (t) >
r vdi ffiQitEL Suy ra co mQt t' E I thoa man x (t') ~ r .
I

I

I

I

Neu ba't d£ng thuc tren khong xay ra vdi mQi t' E I, co t" E I, t":;et'
ma Ix (t") I> r.

39


Md fQng x vdi 1 - tugn hO~lllfa mQt anh x~ lien wc tfen R, ta co th€
gilt sli' ding t" E [t', t'+l].
Bdi tinh lien tt;1c,co mQt khoang md ]ti, t2[ c ] t', t' + 1 [ thoa man
t" E ] ti, t2 [, x(t) I> f, vdi t E ] tl, t2[ va I x(tl) I
I

=

I

X(t2)

I

=r

nhung (2.13) suy ra
I

X(tl)

>

I

I

X(t2)

I

mati thu~n

V~y

I

x(t)

I

::; r, mQi t E I, h~ qua duQc chung minh

H~ qua 2.3' :
Gia sli't6n t~i r > 0 sao cho
(2.10')

(x, f(t, x));::: 0

a.e t E I va mQi x E RDvdi

I

x

I

=r

Thl bai loan (2.1) co it nhfft mQtnghi~m
Chung minh

Ta dinh nghla

K: I x RD-+ RD
f(t,x) ,

get, x)

=

~

ne'u Ix!::;r

r

r

ne'u Ixl;:::r

( I-Ix! ) x+f ( t,!x( J ,

Bdi s1;1'xay d1;1'ngnhu tren, g cling thoa man nhii'ng di~u ki~n
Caratheodory nhuf va trung vdi f tren x ::;r
I

Ne'u a E Ll(I, 14) thoa man

I

f(t, x)

I

I

;:::

- aCt)

a.e t E I va mQi x E RD~vdi

I

x I::;r, thl

(x,g(t,x));::: -a(t{lxI2 + 1), vdi mQi x E RD,a.e tEl

40


Do do di~u ki~n (2.8') cua h~ qua (2.3') thoa man cho g. Ne'u
x E domL thoa man
minlx(t)12 r
tel

Ta co
(2.11')

(x(tJ,g( t,x(tJ)) = (1-lx~t)I)X(tJI2 + ( X(tJ,r(
2

(

1-~

x(t) )

t, Ix~tJI X(t))

J

lx(t)1220

La'y tich phan tren I, di~u ki~n (2.9') cua h~ qua (2.3') thoa man cho
g va bai loan.
(2.12')

X'(t) = g(t,x(t))

,tEl

{ x(o) = x(1)
co it nha't mQt nghi~m.

Ta chung minh nghi~m x nay thoa man x(t) ~ r , tEL Ne'u chung
minh dl1qcdi~u nay, thl nghi~m cua (2.12') cling la nghi~m cua (2.1), di~u
nay se k€t thuc dl1qcchung minh.
I

I

Ne'u x Ia mQt nghi~m cua (2.12'), thl bdi (2.11') ta co a.e tEl
cho x(t) > r.
(2.13 ')

G)( d I dt~x(t)12 =(x(t),g(

t,x(t))) ~ (1-lx~t)I)X(t)12

sao

>0

Do do nghi~m tren khong th~ Ia 1 - tu§n hoan va thoa man x(t) I> r
vdi mQitEL Tdi day chung minh hoan loan tu'dngtv nhl1h~ qua (2.3).
I

H~ qua da dl1qcchung minh.

41


2.3 Dng d~ng cho nhilng ham Vectd.
Ta se cho d day mQtung d\lng cua dinh ly (2.3) de'n phudng phap cua

nhung ham co huang.
Gia sa f thoa man nhung di~u kit%nduQcma ta d phgn dinh ly chinh
Binh nghla 2.1.
V E Cl (Rn,R) duQcgQila mQtham huang (ng~t) cho phudng trlnh
(2.14)

x'

=f(t, x)

ne'u t6n t~i r > 0 saD cho a.e t E I va mQi x E Rll vai

(2.15)

I

x ~ r ta co
I

(V' (x), f(t,x)) ~ 0 « 0).

Chung ta se chung to s\!'t6n t~i cua mQtham co huang thoa man mQt
vai di~u kit%nb6 sung, thl suy ra s\!'t6n t~i cua mQtnghit%m1 - tugn hoan
Tinh cha't 2.1:
Gia sa phudng trlnh (2.14) co mQtham co huang V thoa man
V' (x) :f=0 vai

I

x ~r

I

Va
(2. 16)

Vex) ~ +

00

ne'u x
I

I

~

00

thl bai loan (2. 1) co it nha'"tmQt nghit%m
Chung minh
Bdi nhung di~u kit%ncaratheodory va tinh lien t\lC cua V' t6n t~i
a E L 1(1,R+) saD cho a.e t E I va mQi x vai x ~ r
I

I

Ta co
I

V' (x) 1.1 f(t, x) ~ aCt).

~~tkhac,dung(2.15)
Ta co a.e t E I va mQi x E RD

(V' (x), f(t, x» ~ aCt).
Do do di~u kit%n(i) cua dinh ly (2.2) thoa man, vai nhung anh x~
V va a vila duQcxac dinh nhutren.

42


N6u x E domL thoa man mill x(t) 12 r
I

tEl

dung (2. 15), ta co a.e tEl
(V'(x(t)), f(t,x(t))) ~ 0,
Suy ra f(V'X(t)),f(t,X(t)}lt~O,
I
L1y w(x)

= Vex), suy ra

f(W'(x(t)),f(t,x(t)))dt~O
I

va (V'(x),W'(x))>O,

nhu v~y, di~u ki~n (ii) cua dinh ly (2.2) xay fa, vai r duQc giai thi~u trong

dinh nghla. Do do theo bai tmln (2.3), tinh ch1t duQc chung minh.
Tinh cha't 2.1'. Gia sli' ding phudng trlnh (2. 14) co mQt ham co
huang V thoa man.
V' (x) :;to
0 vai

(2.17) Vex)

~

-

n6u x

00

I

I

~

I

x 2r
I

00

thl bai tmln (2.1) co it nha'tmQtnghi~m

ChUng minh

La'y W(x)

=-

Vex), X ERn, thl W'(X)

=-

V'(X) :;to0, cho mQi x E Rn

~.

VOl X >
- r,
I

I

W(x) ~ +

Va

00

n6u

I

x

I

~

00

(W(X), f (t, X))2 0,

a.e tEl

mQi x E Ril

I

x 12 r lam nhu trong chung minh cua tinh

cha't 2.1.

Ta co

(W' (x), f(t,x)) 2 - aCt),
a.e tEl

mQi x E Ril va a ELI (I, R+), va
f(W'(~(t))f( t,x(t) ))dt 2 0,
I

43

Vai mQix E domL, mill x (t) ~ r. Dung dinh 192.2'
I

I

Chung ta co, bai tmin (2.1) co it nhclt m(>tnghi~m
fJtnh nghia 2.2:
V E C1(Rll, R) gQi la m(>tham g~n co huang cho phuong trlnh
(2.18)

x'

=f(t,

x),

neu t6n t~i r > 0 sao cho, a.e t E I va mQi x E Rfivai

I

x ~ r ta co
I

(V' (x), f(t, x))~ O.

(2. 19)

Tinh cha't 2.2 : Gia stYding phuong trlnh (2.18) co m(>tham g6m co

huang V,thoa man
V' (x) :;t0 vai

I

x ~r
I

Va
(2.20)

Vex) ~

+

00

neu x
I

I

~

00

Thl bai tmin (2.1) co it nhclt m(>tnghi~m
Chung minh

Bdi nhung di~u ki~n Caratheodory va tinh lien we cua V' t6n t~i

a ELI (I, R+) sao cho, a.e t E I, mQi x, vai x :::;r chung ta co,
I

I

V' (x)

1.1

I

f (t, x) ~ - aCt),
I

ngoai ra dung (2.19) (V'(x), f(t, x) ~ 0 chung ta co a.e t E I va mQi x E Rll,
(V' (x), f (t, x)) ~ - aCt)

do do di~u ki~n (i') cua dinh 19(2.2') xay fa. Vai nhung anh x~ V va a. vila
xac dinh d tren.
Bay gio, neu x E domL, thoa man mill x(t) ~ r bdi (2.19), chung ta
I

co, a.e t E I,

(V'(x(t, x(t)) ~ 0
Suy ra

J(V'( x(t)),f( t,x(t) ))dt ~ 0,
I

44

I